Sep 5 
Peter Crooks (Northeastern) 
Integrable systems, Lie algebras, and invariant theory
abstract±
Finitedimensional complex semisimple Lie algebras give a natural setting for studying problems at the interface of symplectic geometry and representation theory. Each such algebra g is naturally a Poisson variety, and by Chevalley's results there are rank(g)many independent Casimirs on g. Mishchenko and Fomenko proved that these Casimirs can be extended to a larger set of independent, pairwiseinvolutive polynomials, which in turn form a completely integrable system on each regular adjoint orbit. Despite being quite classical, these MishchenkoFomenko systems still give rise to a number of interesting questions and new research directions.
I will give a brief overview of MishchenkoFomenko systems and a subsequent description of the questions / research directions that they generate.
Poster.

Sep 12 
Torielli Michele (Hokkaido University) 
On the freeness of hyperplane arrangements
abstract±
An arrangement of hyperplanes is a finite
collection of codimension one affine
subspaces in a finite dimensional vector
space. By definition, an arrangement is free if
and only if its module of logarithmic
derivations is a free module. It turns out that
this notion can be reread in several ways.
In this talk, we will recall the basic notion of
freeness for arrangements, and we will
describe several characterizations of free
arrangements.
Poster.

Sep 19 
Rudy Rodsphon (Northeastern) 
A primer in index theory.
abstract±
In the mid sixties, Atiyah and Singer discovered the index theorem, which relates elliptic PDEs to the topology of the manifold. However, it also has many other connections to other fields, ranging from differential topology to representation theory. The goal of the talk will be to get more familiar with what the index theorem is.
Poster.

Sep 26 
Valeriano Lanza (FAPESP) 
Framed sheaves, ADHM data and quiver varieties
abstract±
After a brief survey on the history of ADHM data, I will focus on Nakajima's description of the moduli space of framed sheaves on P2 as a quiver variety. In particular, in order to undestand the result, some generalities on quiver representations and King's stability conditions will be provided.
Poster.

Oct 3 
Tudor Padurariu (MIT) 
Hall algebras in enumerative geometry
abstract±
A celebrated result of Ringel from the early 90s says that for an ADE quiver, the quantum group of the associated Lie algebra can be constructed using a Hall algebra, which is an algebra that encodes the number of extensions between representations of the quiver over finite fields. The Hall algebras for other categories of dimension 1, such as representations of more general quivers or sheaves on a projective curve, also contain interesting quantum groups. In 2010, Kontsevich and Soibelman constructed a version of Hall algebras for certain categories of dimension 3; for the category of a Calabi Yau 3 fold X, these algebras are related to curve counting theories on X, whenever the algebra can be defined. In this talk, I'll introduce Ringel's and KontsevichSoibelman's constructions, discuss examples of both flavors of Hall algebras, and explain how they are related to counting curves on CalabiYau threefolds.
Poster.

Oct 10 
Rob Silversmith (Northeastern) 
Row echelon form and the Hilbert scheme of points.
abstract±
I will discuss an elementary linear algebra problem that is easy but not interesting, namely to characterize how the rowechelon form of a matrix may change when the columns of the matrix are reversed. Then I will discuss a commutative algebra problem that is interesting but not easy, namely to characterize how the initial ideal of an ideal in C[x,y] may change when x and y are switched. We will see that the second problem is the appropriate generalization of the first problem from vector spaces to rings.
The second question (correctly stated) is open, and I am currently working on it. Geometrically, the problem is about classifying the 1dimensional orbits of the action of a 2torus on the Hilbert scheme of n points in C^2. Combinatorially, the problem is about describing a certain natural family of graphs G_n, where the vertex set of G_n is the set of partitions of n. It also has applications in mirror symmetry and GromovWitten theory.
The talk will be accessible to graduate students.
Poster.

Oct 17 
Bruno Benedetti (Miami) 
Drawing Contractible 2Complexes in R^4
abstract±
From a topological viewpoint, trees are the graphs that are contractible. It is well known that all trees are planar graphs. We will discuss whether the latter fact extends to higher dimensions: For example, can we draw any
contractible complex into a Euclidean space of twice its dimension?
The answer is a very strong “NO!” in dimension two, and a weaker “no” in all other dimensions. The reason comes ultimately algebra: Some presentations of the trivial group are hard to simplify.
This is joint work with Karim Adiprasito.
Poster.

Oct 24 
Olga Turanova (IAS) 
PDEs from evolutionary ecology
abstract±
I will describe the analysis of some PDEs that arise as models of ecological and evolutionary processes. There will be poisonous toads! (Well, there won’t be any actual toads, but at least I will talk about them!)
Poster.

Oct 31 
Brian Williams (Northeastern) 
Spooky factorization algebras and a chiral index theorem
abstract±
Motivated by deformation quantization, NestTsygan and FeiginFelderShoikhet, proved an algebraic version of the AtiyahSinger index theorem. I will sketch how one can recover this result using the language of factorization algebras and the concept of factorization homology. I propose a generalization of the theorem for *vertex algebras* which involves factorization algebras that have a holomorphic flavor.
Poster.

Nov 7 
Alexander Moll (Northeastern) 
Is Randomness Real? Quantum Uncertainty and Chaotic Unpredictability
abstract±
Whereas randomness as formalized by Kolmogorov (1933) is taken as a given in probability theory and statistics, quantum theory and chaos theory are the only two sciences asserting that randomness is a real phenomenon emerging from deterministic dynamical systems. In this talk, we give a leisurely but selfcontained introduction to the Born rule (1926), an account of quantum randomness formalized by von Neumann (1932) without Feynman’s path integral (1948), with an emphasis on the role of commutativity in defining joint correlation functions and quantum integrability. By the end, we will be able to prove that the random value of the position operator in a coherent state is a Gaussian random variable, a simple but fundamental mathematical fact measured experimentally by BreitenbachSchillerMlynek (1997).
Poster.

Nov 14 
Thomas Koberda (UVA) 
Group actions and regularity
abstract±
I'll talk about groups acting on the circle and on the interval. It turns out that groups which can act are highly constrained by regularity, which is to say the level of differentiability of the action. I'll try and give an intuitive understanding of what distinguishes actions which are, say 99 times differentiable versus 100 times differentiable.
Poster.

Nov 21 
No Seminar 
Thanksgiving Break

Nov 28 
Ignacio Barros (Northeastern) 
Hopes, conjectures and frustrations regarding M_g
abstract±
We will discuss a classical problem in curve theory; the description of the birational geometry of the moduli space of curves. Basic invariants, open cases and relations to other moduli spaces, such as the moduli of K3 surfaces. This will be a partial report on a joint work with S. Mullane.
Poster.

Dec 5 




End of Semester (Seminar Resumes Spring 2019) 